LMFDB - Embedded newform 153.4.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [153,4,Mod(1,153)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("153.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(153, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	153.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,20,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$9.02729223088$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	153.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.24264 q^{2} +10.0000 q^{4} +13.9706 q^{5} +12.9706 q^{7} +8.48528 q^{8} +59.2721 q^{10} -49.9706 q^{11} +32.9411 q^{13} +55.0294 q^{14} -44.0000 q^{16} +17.0000 q^{17} +54.8823 q^{19} +139.706 q^{20} -212.007 q^{22} +82.0294 q^{23} +70.1766 q^{25} +139.757 q^{26} +129.706 q^{28} -289.882 q^{29} +232.059 q^{31} -254.558 q^{32} +72.1249 q^{34} +181.206 q^{35} -227.529 q^{37} +232.846 q^{38} +118.544 q^{40} -437.735 q^{41} -158.882 q^{43} -499.706 q^{44} +348.021 q^{46} -159.088 q^{47} -174.765 q^{49} +297.734 q^{50} +329.411 q^{52} -376.087 q^{53} -698.117 q^{55} +110.059 q^{56} -1229.87 q^{58} +185.294 q^{59} +861.852 q^{61} +984.542 q^{62} -728.000 q^{64} +460.206 q^{65} -178.530 q^{67} +170.000 q^{68} +768.792 q^{70} +1161.59 q^{71} +383.088 q^{73} -965.324 q^{74} +548.823 q^{76} -648.146 q^{77} +254.000 q^{79} -614.705 q^{80} -1857.15 q^{82} -447.088 q^{83} +237.500 q^{85} -674.080 q^{86} -424.014 q^{88} +1213.15 q^{89} +427.265 q^{91} +820.294 q^{92} -674.955 q^{94} +766.736 q^{95} +291.383 q^{97} -741.463 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{4} - 6 q^{5} - 8 q^{7} + 144 q^{10} - 66 q^{11} - 2 q^{13} + 144 q^{14} - 88 q^{16} + 34 q^{17} - 26 q^{19} - 60 q^{20} - 144 q^{22} + 198 q^{23} + 344 q^{25} + 288 q^{26} - 80 q^{28} - 444 q^{29}+ \cdots - 1152 q^{98}+O(q^{100}) $ 2 * q + 20 * q^4 - 6 * q^5 - 8 * q^7 + 144 * q^10 - 66 * q^11 - 2 * q^13 + 144 * q^14 - 88 * q^16 + 34 * q^17 - 26 * q^19 - 60 * q^20 - 144 * q^22 + 198 * q^23 + 344 * q^25 + 288 * q^26 - 80 * q^28 - 444 * q^29 + 532 * q^31 + 600 * q^35 + 88 * q^37 + 576 * q^38 + 288 * q^40 - 570 * q^41 - 182 * q^43 - 660 * q^44 - 144 * q^46 - 420 * q^47 - 78 * q^49 - 864 * q^50 - 20 * q^52 + 300 * q^53 - 378 * q^55 + 288 * q^56 - 576 * q^58 - 444 * q^59 + 400 * q^61 - 288 * q^62 - 1456 * q^64 + 1158 * q^65 - 968 * q^67 + 340 * q^68 - 1008 * q^70 + 1848 * q^71 + 868 * q^73 - 2304 * q^74 - 260 * q^76 - 312 * q^77 + 508 * q^79 + 264 * q^80 - 1296 * q^82 - 996 * q^83 - 102 * q^85 - 576 * q^86 - 288 * q^88 + 288 * q^89 + 1160 * q^91 + 1980 * q^92 + 432 * q^94 + 2382 * q^95 + 1024 * q^97 - 1152 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.24264	1.50000	0.750000	−	0.661438i	$-0.230053\pi$
$2$	4.24264	1.50000	0.750000	+	0.661438i	$0.230053\pi$
$3$	0	0
$3$	0	0
$4$	10.0000	1.25000
$5$	13.9706	1.24957	0.624783	−	0.780799i	$-0.285188\pi$
$5$	13.9706	1.24957	0.624783	+	0.780799i	$0.285188\pi$
$6$	0	0
$7$	12.9706	0.700345	0.350172	−	0.936685i	$-0.386123\pi$
$7$	12.9706	0.700345	0.350172	+	0.936685i	$0.386123\pi$
$8$	8.48528	0.375000
$9$	0	0
$10$	59.2721	1.87435
$11$	−49.9706	−1.36970	−0.684850	−	0.728684i	$-0.740132\pi$
$11$	−49.9706	−1.36970	−0.684850	+	0.728684i	$0.740132\pi$
$12$	0	0
$13$	32.9411	0.702786	0.351393	−	0.936228i	$-0.385708\pi$
$13$	32.9411	0.702786	0.351393	+	0.936228i	$0.385708\pi$
$14$	55.0294	1.05052
$15$	0	0
$16$	−44.0000	−0.687500
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	54.8823	0.662676	0.331338	−	0.943512i	$-0.392500\pi$
$19$	54.8823	0.662676	0.331338	+	0.943512i	$0.392500\pi$
$20$	139.706	1.56196
$21$	0	0
$22$	−212.007	−2.05455
$23$	82.0294	0.743666	0.371833	−	0.928300i	$-0.378729\pi$
$23$	82.0294	0.743666	0.371833	+	0.928300i	$0.378729\pi$
$24$	0	0
$25$	70.1766	0.561413
$26$	139.757	1.05418
$27$	0	0
$28$	129.706	0.875431
$29$	−289.882	−1.85620	−0.928100	−	0.372332i	$-0.878558\pi$
$29$	−289.882	−1.85620	−0.928100	+	0.372332i	$0.878558\pi$
$30$	0	0
$31$	232.059	1.34448	0.672242	−	0.740331i	$-0.265331\pi$
$31$	232.059	1.34448	0.672242	+	0.740331i	$0.265331\pi$
$32$	−254.558	−1.40625
$33$	0	0
$34$	72.1249	0.363803
$35$	181.206	0.875126
$36$	0	0
$37$	−227.529	−1.01096	−0.505480	−	0.862838i	$-0.668685\pi$
$37$	−227.529	−1.01096	−0.505480	+	0.862838i	$0.668685\pi$
$38$	232.846	0.994015
$39$	0	0
$40$	118.544	0.468587
$41$	−437.735	−1.66738	−0.833692	−	0.552230i	$-0.813777\pi$
$41$	−437.735	−1.66738	−0.833692	+	0.552230i	$0.813777\pi$
$42$	0	0
$43$	−158.882	−0.563472	−0.281736	−	0.959492i	$-0.590910\pi$
$43$	−158.882	−0.563472	−0.281736	+	0.959492i	$0.590910\pi$
$44$	−499.706	−1.71212
$45$	0	0
$46$	348.021	1.11550
$47$	−159.088	−0.493732	−0.246866	−	0.969050i	$-0.579401\pi$
$47$	−159.088	−0.493732	−0.246866	+	0.969050i	$0.579401\pi$
$48$	0	0
$49$	−174.765	−0.509517
$50$	297.734	0.842119
$51$	0	0
$52$	329.411	0.878483
$53$	−376.087	−0.974709	−0.487355	−	0.873204i	$-0.662038\pi$
$53$	−376.087	−0.974709	−0.487355	+	0.873204i	$0.662038\pi$
$54$	0	0
$55$	−698.117	−1.71153
$56$	110.059	0.262629
$57$	0	0
$58$	−1229.87	−2.78430
$59$	185.294	0.408867	0.204434	−	0.978880i	$-0.434465\pi$
$59$	185.294	0.408867	0.204434	+	0.978880i	$0.434465\pi$
$60$	0	0
$61$	861.852	1.80900	0.904499	−	0.426477i	$-0.140245\pi$
$61$	861.852	1.80900	0.904499	+	0.426477i	$0.140245\pi$
$62$	984.542	2.01673
$63$	0	0
$64$	−728.000	−1.42188
$65$	460.206	0.878177
$66$	0	0
$67$	−178.530	−0.325536	−0.162768	−	0.986664i	$-0.552042\pi$
$67$	−178.530	−0.325536	−0.162768	+	0.986664i	$0.552042\pi$
$68$	170.000	0.303170
$69$	0	0
$70$	768.792	1.31269
$71$	1161.59	1.94162	0.970811	−	0.239847i	$-0.0770973\pi$
$71$	1161.59	1.94162	0.970811	+	0.239847i	$0.0770973\pi$
$72$	0	0
$73$	383.088	0.614207	0.307103	−	0.951676i	$-0.400640\pi$
$73$	383.088	0.614207	0.307103	+	0.951676i	$0.400640\pi$
$74$	−965.324	−1.51644
$75$	0	0
$76$	548.823	0.828346
$77$	−648.146	−0.959261
$78$	0	0
$79$	254.000	0.361737	0.180869	−	0.983507i	$-0.442109\pi$
$79$	254.000	0.361737	0.180869	+	0.983507i	$0.442109\pi$
$80$	−614.705	−0.859076
$81$	0	0
$82$	−1857.15	−2.50108
$83$	−447.088	−0.591257	−0.295628	−	0.955303i	$-0.595529\pi$
$83$	−447.088	−0.591257	−0.295628	+	0.955303i	$0.595529\pi$
$84$	0	0
$85$	237.500	0.303064
$86$	−674.080	−0.845209
$87$	0	0
$88$	−424.014	−0.513637
$89$	1213.15	1.44487	0.722433	−	0.691440i	$-0.243024\pi$
$89$	1213.15	1.44487	0.722433	+	0.691440i	$0.243024\pi$
$90$	0	0
$91$	427.265	0.492193
$92$	820.294	0.929583
$93$	0	0
$94$	−674.955	−0.740598
$95$	766.736	0.828057
$96$	0	0
$97$	291.383	0.305004	0.152502	−	0.988303i	$-0.451267\pi$
$97$	291.383	0.305004	0.152502	+	0.988303i	$0.451267\pi$
$98$	−741.463	−0.764276
$99$	0	0
$100$	701.766	0.701766
$101$	−1568.53	−1.54529	−0.772645	−	0.634838i	$-0.781067\pi$
$101$	−1568.53	−1.54529	−0.772645	+	0.634838i	$0.781067\pi$
$102$	0	0
$103$	412.647	0.394750	0.197375	−	0.980328i	$-0.436758\pi$
$103$	412.647	0.394750	0.197375	+	0.980328i	$0.436758\pi$
$104$	279.515	0.263545
$105$	0	0
$106$	−1595.60	−1.46206
$107$	239.382	0.216280	0.108140	−	0.994136i	$-0.465511\pi$
$107$	239.382	0.216280	0.108140	+	0.994136i	$0.465511\pi$
$108$	0	0
$109$	−759.647	−0.667532	−0.333766	−	0.942656i	$-0.608319\pi$
$109$	−759.647	−0.667532	−0.333766	+	0.942656i	$0.608319\pi$
$110$	−2961.86	−2.56729
$111$	0	0
$112$	−570.705	−0.481487
$113$	292.383	0.243408	0.121704	−	0.992566i	$-0.461164\pi$
$113$	292.383	0.243408	0.121704	+	0.992566i	$0.461164\pi$
$114$	0	0
$115$	1146.00	0.929259
$116$	−2898.82	−2.32025
$117$	0	0
$118$	786.134	0.613301
$119$	220.500	0.169859
$120$	0	0
$121$	1166.06	0.876076
$122$	3656.53	2.71350
$123$	0	0
$124$	2320.59	1.68061
$125$	−765.913	−0.548043
$126$	0	0
$127$	−2646.41	−1.84906	−0.924531	−	0.381107i	$-0.875543\pi$
$127$	−2646.41	−1.84906	−0.924531	+	0.381107i	$0.875543\pi$
$128$	−1052.17	−0.726562
$129$	0	0
$130$	1952.49	1.31727
$131$	1964.15	1.30999	0.654994	−	0.755634i	$-0.272671\pi$
$131$	1964.15	1.30999	0.654994	+	0.755634i	$0.272671\pi$
$132$	0	0
$133$	711.854	0.464102
$134$	−757.438	−0.488304
$135$	0	0
$136$	144.250	0.0909509
$137$	2083.15	1.29909	0.649544	−	0.760324i	$-0.274960\pi$
$137$	2083.15	1.29909	0.649544	+	0.760324i	$0.274960\pi$
$138$	0	0
$139$	1715.73	1.04695	0.523477	−	0.852040i	$-0.324635\pi$
$139$	1715.73	1.04695	0.523477	+	0.852040i	$0.324635\pi$
$140$	1812.06	1.09391
$141$	0	0
$142$	4928.20	2.91243
$143$	−1646.09	−0.962606
$144$	0	0
$145$	−4049.82	−2.31944
$146$	1625.31	0.921310
$147$	0	0
$148$	−2275.29	−1.26370
$149$	1679.62	0.923487	0.461743	−	0.887014i	$-0.347224\pi$
$149$	1679.62	0.923487	0.461743	+	0.887014i	$0.347224\pi$
$150$	0	0
$151$	644.353	0.347263	0.173632	−	0.984811i	$-0.444450\pi$
$151$	644.353	0.347263	0.173632	+	0.984811i	$0.444450\pi$
$152$	465.691	0.248504
$153$	0	0
$154$	−2749.85	−1.43889
$155$	3241.99	1.68002
$156$	0	0
$157$	2130.53	1.08302	0.541512	−	0.840693i	$-0.317852\pi$
$157$	2130.53	1.08302	0.541512	+	0.840693i	$0.317852\pi$
$158$	1077.63	0.542606
$159$	0	0
$160$	−3556.32	−1.75720
$161$	1063.97	0.520822
$162$	0	0
$163$	3582.41	1.72145	0.860724	−	0.509072i	$-0.170011\pi$
$163$	3582.41	1.72145	0.860724	+	0.509072i	$0.170011\pi$
$164$	−4377.35	−2.08423
$165$	0	0
$166$	−1896.84	−0.886885
$167$	1861.79	0.862694	0.431347	−	0.902186i	$-0.358038\pi$
$167$	1861.79	0.862694	0.431347	+	0.902186i	$0.358038\pi$
$168$	0	0
$169$	−1111.88	−0.506091
$170$	1007.63	0.454596
$171$	0	0
$172$	−1588.82	−0.704341
$173$	−2612.50	−1.14812	−0.574059	−	0.818814i	$-0.694632\pi$
$173$	−2612.50	−1.14812	−0.574059	+	0.818814i	$0.694632\pi$
$174$	0	0
$175$	910.230	0.393183
$176$	2198.70	0.941668
$177$	0	0
$178$	5146.94	2.16730
$179$	−126.646	−0.0528824	−0.0264412	−	0.999650i	$-0.508417\pi$
$179$	−126.646	−0.0528824	−0.0264412	+	0.999650i	$0.508417\pi$
$180$	0	0
$181$	1783.26	0.732314	0.366157	−	0.930553i	$-0.380673\pi$
$181$	1783.26	0.732314	0.366157	+	0.930553i	$0.380673\pi$
$182$	1812.73	0.738289
$183$	0	0
$184$	696.043	0.278875
$185$	−3178.71	−1.26326
$186$	0	0
$187$	−849.500	−0.332201
$188$	−1590.88	−0.617165
$189$	0	0
$190$	3252.99	1.24209
$191$	−2144.44	−0.812388	−0.406194	−	0.913787i	$-0.633144\pi$
$191$	−2144.44	−0.812388	−0.406194	+	0.913787i	$0.633144\pi$
$192$	0	0
$193$	3205.82	1.19565	0.597823	−	0.801628i	$-0.296032\pi$
$193$	3205.82	1.19565	0.597823	+	0.801628i	$0.296032\pi$
$194$	1236.23	0.457507
$195$	0	0
$196$	−1747.65	−0.636897
$197$	−2768.32	−1.00119	−0.500596	−	0.865681i	$-0.666886\pi$
$197$	−2768.32	−1.00119	−0.500596	+	0.865681i	$0.666886\pi$
$198$	0	0
$199$	−712.792	−0.253912	−0.126956	−	0.991908i	$-0.540521\pi$
$199$	−712.792	−0.253912	−0.126956	+	0.991908i	$0.540521\pi$
$200$	595.468	0.210530
$201$	0	0
$202$	−6654.70	−2.31794
$203$	−3759.94	−1.29998
$204$	0	0
$205$	−6115.41	−2.08350
$206$	1750.71	0.592126
$207$	0	0
$208$	−1449.41	−0.483166
$209$	−2742.50	−0.907667
$210$	0	0
$211$	787.591	0.256967	0.128483	−	0.991712i	$-0.458989\pi$
$211$	787.591	0.256967	0.128483	+	0.991712i	$0.458989\pi$
$212$	−3760.87	−1.21839
$213$	0	0
$214$	1015.61	0.324419
$215$	−2219.67	−0.704096
$216$	0	0
$217$	3009.93	0.941602
$218$	−3222.91	−1.00130
$219$	0	0
$220$	−6981.17	−2.13941
$221$	559.999	0.170451
$222$	0	0
$223$	−2926.53	−0.878811	−0.439405	−	0.898289i	$-0.644811\pi$
$223$	−2926.53	−0.878811	−0.439405	+	0.898289i	$0.644811\pi$
$224$	−3301.77	−0.984860
$225$	0	0
$226$	1240.47	0.365111
$227$	6212.03	1.81633	0.908164	−	0.418614i	$-0.137484\pi$
$227$	6212.03	1.81633	0.908164	+	0.418614i	$0.137484\pi$
$228$	0	0
$229$	−4516.35	−1.30327	−0.651635	−	0.758533i	$-0.725916\pi$
$229$	−4516.35	−1.30327	−0.651635	+	0.758533i	$0.725916\pi$
$230$	4862.06	1.39389
$231$	0	0
$232$	−2459.73	−0.696075
$233$	−3547.26	−0.997376	−0.498688	−	0.866782i	$-0.666185\pi$
$233$	−3547.26	−0.997376	−0.498688	+	0.866782i	$0.666185\pi$
$234$	0	0
$235$	−2222.55	−0.616951
$236$	1852.94	0.511084
$237$	0	0
$238$	935.500	0.254788
$239$	−726.969	−0.196752	−0.0983760	−	0.995149i	$-0.531365\pi$
$239$	−726.969	−0.196752	−0.0983760	+	0.995149i	$0.531365\pi$
$240$	0	0
$241$	1689.67	0.451623	0.225812	−	0.974171i	$-0.427497\pi$
$241$	1689.67	0.451623	0.225812	+	0.974171i	$0.427497\pi$
$242$	4947.16	1.31411
$243$	0	0
$244$	8618.52	2.26125
$245$	−2441.56	−0.636675
$246$	0	0
$247$	1807.88	0.465720
$248$	1969.08	0.504182
$249$	0	0
$250$	−3249.50	−0.822065
$251$	−911.707	−0.229269	−0.114634	−	0.993408i	$-0.536570\pi$
$251$	−911.707	−0.229269	−0.114634	+	0.993408i	$0.536570\pi$
$252$	0	0
$253$	−4099.06	−1.01860
$254$	−11227.8	−2.77359
$255$	0	0
$256$	1360.00	0.332031
$257$	3123.56	0.758141	0.379070	−	0.925368i	$-0.376244\pi$
$257$	3123.56	0.758141	0.379070	+	0.925368i	$0.376244\pi$
$258$	0	0
$259$	−2951.18	−0.708021
$260$	4602.06	1.09772
$261$	0	0
$262$	8333.17	1.96498
$263$	−137.288	−0.0321884	−0.0160942	−	0.999870i	$-0.505123\pi$
$263$	−137.288	−0.0321884	−0.0160942	+	0.999870i	$0.505123\pi$
$264$	0	0
$265$	−5254.15	−1.21796
$266$	3020.14	0.696153
$267$	0	0
$268$	−1785.30	−0.406920
$269$	2030.26	0.460175	0.230087	−	0.973170i	$-0.426099\pi$
$269$	2030.26	0.460175	0.230087	+	0.973170i	$0.426099\pi$
$270$	0	0
$271$	−1187.23	−0.266123	−0.133061	−	0.991108i	$-0.542481\pi$
$271$	−1187.23	−0.266123	−0.133061	+	0.991108i	$0.542481\pi$
$272$	−748.000	−0.166743
$273$	0	0
$274$	8838.04	1.94863
$275$	−3506.77	−0.768967
$276$	0	0
$277$	3027.91	0.656786	0.328393	−	0.944541i	$-0.393493\pi$
$277$	3027.91	0.656786	0.328393	+	0.944541i	$0.393493\pi$
$278$	7279.24	1.57043
$279$	0	0
$280$	1537.58	0.328172
$281$	5519.90	1.17185	0.585925	−	0.810365i	$-0.300731\pi$
$281$	5519.90	1.17185	0.585925	+	0.810365i	$0.300731\pi$
$282$	0	0
$283$	−5888.17	−1.23680	−0.618402	−	0.785862i	$-0.712220\pi$
$283$	−5888.17	−1.23680	−0.618402	+	0.785862i	$0.712220\pi$
$284$	11615.9	2.42703
$285$	0	0
$286$	−6983.75	−1.44391
$287$	−5677.67	−1.16774
$288$	0	0
$289$	289.000	0.0588235
$290$	−17181.9	−3.47916
$291$	0	0
$292$	3830.88	0.767758
$293$	2873.06	0.572852	0.286426	−	0.958102i	$-0.407533\pi$
$293$	2873.06	0.572852	0.286426	+	0.958102i	$0.407533\pi$
$294$	0	0
$295$	2588.65	0.510906
$296$	−1930.65	−0.379110
$297$	0	0
$298$	7126.01	1.38523
$299$	2702.14	0.522638
$300$	0	0
$301$	−2060.79	−0.394625
$302$	2733.76	0.520895
$303$	0	0
$304$	−2414.82	−0.455590
$305$	12040.6	2.26046
$306$	0	0
$307$	318.234	0.0591614	0.0295807	−	0.999562i	$-0.490583\pi$
$307$	318.234	0.0591614	0.0295807	+	0.999562i	$0.490583\pi$
$308$	−6481.46	−1.19908
$309$	0	0
$310$	13754.6	2.52003
$311$	1940.05	0.353731	0.176866	−	0.984235i	$-0.443404\pi$
$311$	1940.05	0.353731	0.176866	+	0.984235i	$0.443404\pi$
$312$	0	0
$313$	−5487.84	−0.991026	−0.495513	−	0.868600i	$-0.665020\pi$
$313$	−5487.84	−0.991026	−0.495513	+	0.868600i	$0.665020\pi$
$314$	9039.07	1.62454
$315$	0	0
$316$	2540.00	0.452171
$317$	−1337.29	−0.236940	−0.118470	−	0.992958i	$-0.537799\pi$
$317$	−1337.29	−0.236940	−0.118470	+	0.992958i	$0.537799\pi$
$318$	0	0
$319$	14485.6	2.54243
$320$	−10170.6	−1.77673
$321$	0	0
$322$	4514.03	0.781234
$323$	932.998	0.160723
$324$	0	0
$325$	2311.70	0.394553
$326$	15198.9	2.58217
$327$	0	0
$328$	−3714.31	−0.625269
$329$	−2063.46	−0.345783
$330$	0	0
$331$	−4826.46	−0.801470	−0.400735	−	0.916194i	$-0.631245\pi$
$331$	−4826.46	−0.801470	−0.400735	+	0.916194i	$0.631245\pi$
$332$	−4470.88	−0.739071
$333$	0	0
$334$	7898.92	1.29404
$335$	−2494.16	−0.406778
$336$	0	0
$337$	−8265.21	−1.33601	−0.668004	−	0.744158i	$-0.732851\pi$
$337$	−8265.21	−1.33601	−0.668004	+	0.744158i	$0.732851\pi$
$338$	−4717.32	−0.759137
$339$	0	0
$340$	2375.00	0.378830
$341$	−11596.1	−1.84154
$342$	0	0
$343$	−6715.70	−1.05718
$344$	−1348.16	−0.211302
$345$	0	0
$346$	−11083.9	−1.72218
$347$	−5841.35	−0.903688	−0.451844	−	0.892097i	$-0.649234\pi$
$347$	−5841.35	−0.903688	−0.451844	+	0.892097i	$0.649234\pi$
$348$	0	0
$349$	−3873.11	−0.594048	−0.297024	−	0.954870i	$-0.595994\pi$
$349$	−3873.11	−0.594048	−0.297024	+	0.954870i	$0.595994\pi$
$350$	3861.78	0.589774
$351$	0	0
$352$	12720.4	1.92614
$353$	−4020.29	−0.606171	−0.303085	−	0.952963i	$-0.598017\pi$
$353$	−4020.29	−0.606171	−0.303085	+	0.952963i	$0.598017\pi$
$354$	0	0
$355$	16228.0	2.42618
$356$	12131.5	1.80608
$357$	0	0
$358$	−537.313	−0.0793237
$359$	2272.06	0.334024	0.167012	−	0.985955i	$-0.446588\pi$
$359$	2272.06	0.334024	0.167012	+	0.985955i	$0.446588\pi$
$360$	0	0
$361$	−3846.94	−0.560860
$362$	7565.75	1.09847
$363$	0	0
$364$	4272.65	0.615241
$365$	5351.96	0.767491
$366$	0	0
$367$	7353.58	1.04592	0.522962	−	0.852356i	$-0.324827\pi$
$367$	7353.58	1.04592	0.522962	+	0.852356i	$0.324827\pi$
$368$	−3609.30	−0.511270
$369$	0	0
$370$	−13486.1	−1.89489
$371$	−4878.07	−0.682632
$372$	0	0
$373$	320.701	0.0445182	0.0222591	−	0.999752i	$-0.492914\pi$
$373$	320.701	0.0445182	0.0222591	+	0.999752i	$0.492914\pi$
$374$	−3604.12	−0.498301
$375$	0	0
$376$	−1349.91	−0.185150
$377$	−9549.05	−1.30451
$378$	0	0
$379$	−700.903	−0.0949946	−0.0474973	−	0.998871i	$-0.515125\pi$
$379$	−700.903	−0.0949946	−0.0474973	+	0.998871i	$0.515125\pi$
$380$	7667.36	1.03507
$381$	0	0
$382$	−9098.08	−1.21858
$383$	−8217.97	−1.09639	−0.548196	−	0.836350i	$-0.684685\pi$
$383$	−8217.97	−1.09639	−0.548196	+	0.836350i	$0.684685\pi$
$384$	0	0
$385$	−9054.97	−1.19866
$386$	13601.1	1.79347
$387$	0	0
$388$	2913.83	0.381256
$389$	3800.59	0.495366	0.247683	−	0.968841i	$-0.420331\pi$
$389$	3800.59	0.495366	0.247683	+	0.968841i	$0.420331\pi$
$390$	0	0
$391$	1394.50	0.180366
$392$	−1482.93	−0.191069
$393$	0	0
$394$	−11745.0	−1.50179
$395$	3548.52	0.452014
$396$	0	0
$397$	−5955.38	−0.752876	−0.376438	−	0.926442i	$-0.622851\pi$
$397$	−5955.38	−0.752876	−0.376438	+	0.926442i	$0.622851\pi$
$398$	−3024.12	−0.380868
$399$	0	0
$400$	−3087.77	−0.385971
$401$	−581.967	−0.0724739	−0.0362370	−	0.999343i	$-0.511537\pi$
$401$	−581.967	−0.0724739	−0.0362370	+	0.999343i	$0.511537\pi$
$402$	0	0
$403$	7644.28	0.944885
$404$	−15685.3	−1.93161
$405$	0	0
$406$	−15952.1	−1.94997
$407$	11369.8	1.38471
$408$	0	0
$409$	−11357.7	−1.37311	−0.686555	−	0.727078i	$-0.740878\pi$
$409$	−11357.7	−1.37311	−0.686555	+	0.727078i	$0.740878\pi$
$410$	−25945.5	−3.12526
$411$	0	0
$412$	4126.47	0.493438
$413$	2403.36	0.286348
$414$	0	0
$415$	−6246.08	−0.738814
$416$	−8385.44	−0.988294
$417$	0	0
$418$	−11635.4	−1.36150
$419$	20.9420	0.00244173	0.00122086	−	0.999999i	$-0.499611\pi$
$419$	20.9420	0.00244173	0.00122086	+	0.999999i	$0.499611\pi$
$420$	0	0
$421$	4455.28	0.515765	0.257882	−	0.966176i	$-0.416975\pi$
$421$	4455.28	0.515765	0.257882	+	0.966176i	$0.416975\pi$
$422$	3341.47	0.385450
$423$	0	0
$424$	−3191.21	−0.365516
$425$	1193.00	0.136163
$426$	0	0
$427$	11178.7	1.26692
$428$	2393.82	0.270349
$429$	0	0
$430$	−9417.28	−1.05614
$431$	2410.99	0.269451	0.134725	−	0.990883i	$-0.456985\pi$
$431$	2410.99	0.269451	0.134725	+	0.990883i	$0.456985\pi$
$432$	0	0
$433$	−1653.71	−0.183539	−0.0917693	−	0.995780i	$-0.529252\pi$
$433$	−1653.71	−0.183539	−0.0917693	+	0.995780i	$0.529252\pi$
$434$	12770.1	1.41240
$435$	0	0
$436$	−7596.47	−0.834415
$437$	4501.96	0.492810
$438$	0	0
$439$	14852.5	1.61474	0.807371	−	0.590044i	$-0.200890\pi$
$439$	14852.5	1.61474	0.807371	+	0.590044i	$0.200890\pi$
$440$	−5923.72	−0.641823
$441$	0	0
$442$	2375.88	0.255676
$443$	5393.26	0.578423	0.289212	−	0.957265i	$-0.406607\pi$
$443$	5393.26	0.578423	0.289212	+	0.957265i	$0.406607\pi$
$444$	0	0
$445$	16948.3	1.80546
$446$	−12416.2	−1.31822
$447$	0	0
$448$	−9442.57	−0.995802
$449$	−6144.82	−0.645862	−0.322931	−	0.946422i	$-0.604668\pi$
$449$	−6144.82	−0.645862	−0.322931	+	0.946422i	$0.604668\pi$
$450$	0	0
$451$	21873.9	2.28381
$452$	2923.83	0.304259
$453$	0	0
$454$	26355.4	2.72449
$455$	5969.13	0.615027
$456$	0	0
$457$	6041.18	0.618369	0.309184	−	0.951002i	$-0.399944\pi$
$457$	6041.18	0.618369	0.309184	+	0.951002i	$0.399944\pi$
$458$	−19161.2	−1.95490
$459$	0	0
$460$	11460.0	1.16157
$461$	−13829.8	−1.39722	−0.698610	−	0.715503i	$-0.746198\pi$
$461$	−13829.8	−1.39722	−0.698610	+	0.715503i	$0.746198\pi$
$462$	0	0
$463$	12585.3	1.26326	0.631629	−	0.775271i	$-0.282387\pi$
$463$	12585.3	1.26326	0.631629	+	0.775271i	$0.282387\pi$
$464$	12754.8	1.27614
$465$	0	0
$466$	−15049.7	−1.49606
$467$	6561.91	0.650212	0.325106	−	0.945678i	$-0.394600\pi$
$467$	6561.91	0.650212	0.325106	+	0.945678i	$0.394600\pi$
$468$	0	0
$469$	−2315.63	−0.227987
$470$	−9429.49	−0.925426
$471$	0	0
$472$	1572.27	0.153325
$473$	7939.44	0.771788
$474$	0	0
$475$	3851.45	0.372035
$476$	2205.00	0.212323
$477$	0	0
$478$	−3084.27	−0.295128
$479$	16609.4	1.58435	0.792175	−	0.610294i	$-0.208949\pi$
$479$	16609.4	1.58435	0.792175	+	0.610294i	$0.208949\pi$
$480$	0	0
$481$	−7495.06	−0.710489
$482$	7168.66	0.677435
$483$	0	0
$484$	11660.6	1.09509
$485$	4070.78	0.381123
$486$	0	0
$487$	−7999.46	−0.744333	−0.372166	−	0.928166i	$-0.621385\pi$
$487$	−7999.46	−0.744333	−0.372166	+	0.928166i	$0.621385\pi$
$488$	7313.06	0.678374
$489$	0	0
$490$	−10358.7	−0.955013
$491$	11669.1	1.07255	0.536274	−	0.844044i	$-0.319831\pi$
$491$	11669.1	1.07255	0.536274	+	0.844044i	$0.319831\pi$
$492$	0	0
$493$	−4928.00	−0.450194
$494$	7670.20	0.698580
$495$	0	0
$496$	−10210.6	−0.924333
$497$	15066.4	1.35980
$498$	0	0
$499$	−20896.2	−1.87464	−0.937319	−	0.348473i	$-0.886700\pi$
$499$	−20896.2	−1.87464	−0.937319	+	0.348473i	$0.886700\pi$
$500$	−7659.13	−0.685054
$501$	0	0
$502$	−3868.05	−0.343903
$503$	−17429.1	−1.54498	−0.772490	−	0.635028i	$-0.780989\pi$
$503$	−17429.1	−1.54498	−0.772490	+	0.635028i	$0.780989\pi$
$504$	0	0
$505$	−21913.2	−1.93094
$506$	−17390.8	−1.52790
$507$	0	0
$508$	−26464.1	−2.31133
$509$	−1020.29	−0.0888481	−0.0444240	−	0.999013i	$-0.514145\pi$
$509$	−1020.29	−0.0888481	−0.0444240	+	0.999013i	$0.514145\pi$
$510$	0	0
$511$	4968.87	0.430156
$512$	14187.4	1.22461
$513$	0	0
$514$	13252.1	1.13721
$515$	5764.91	0.493266
$516$	0	0
$517$	7949.73	0.676265
$518$	−12520.8	−1.06203
$519$	0	0
$520$	3904.98	0.329317
$521$	5281.92	0.444155	0.222078	−	0.975029i	$-0.428716\pi$
$521$	5281.92	0.444155	0.222078	+	0.975029i	$0.428716\pi$
$522$	0	0
$523$	−15906.1	−1.32988	−0.664938	−	0.746898i	$-0.731542\pi$
$523$	−15906.1	−1.32988	−0.664938	+	0.746898i	$0.731542\pi$
$524$	19641.5	1.63748
$525$	0	0
$526$	−582.465	−0.0482827
$527$	3945.00	0.326085
$528$	0	0
$529$	−5438.17	−0.446961
$530$	−22291.5	−1.82694
$531$	0	0
$532$	7118.54	0.580127
$533$	−14419.5	−1.17181
$534$	0	0
$535$	3344.30	0.270255
$536$	−1514.88	−0.122076
$537$	0	0
$538$	8613.66	0.690262
$539$	8733.08	0.697886
$540$	0	0
$541$	21923.7	1.74228	0.871139	−	0.491036i	$-0.163382\pi$
$541$	21923.7	1.74228	0.871139	+	0.491036i	$0.163382\pi$
$542$	−5037.01	−0.399184
$543$	0	0
$544$	−4327.49	−0.341066
$545$	−10612.7	−0.834124
$546$	0	0
$547$	4960.74	0.387762	0.193881	−	0.981025i	$-0.437892\pi$
$547$	4960.74	0.387762	0.193881	+	0.981025i	$0.437892\pi$
$548$	20831.5	1.62386
$549$	0	0
$550$	−14877.9	−1.15345
$551$	−15909.4	−1.23006
$552$	0	0
$553$	3294.52	0.253341
$554$	12846.3	0.985179
$555$	0	0
$556$	17157.3	1.30869
$557$	−22404.1	−1.70429	−0.852146	−	0.523305i	$-0.824699\pi$
$557$	−22404.1	−1.70429	−0.852146	+	0.523305i	$0.824699\pi$
$558$	0	0
$559$	−5233.76	−0.396001
$560$	−7973.07	−0.601649
$561$	0	0
$562$	23419.0	1.75778
$563$	17361.5	1.29965	0.649824	−	0.760085i	$-0.274843\pi$
$563$	17361.5	1.29965	0.649824	+	0.760085i	$0.274843\pi$
$564$	0	0
$565$	4084.75	0.304154
$566$	−24981.4	−1.85521
$567$	0	0
$568$	9856.40	0.728108
$569$	−1980.78	−0.145938	−0.0729690	−	0.997334i	$-0.523247\pi$
$569$	−1980.78	−0.145938	−0.0729690	+	0.997334i	$0.523247\pi$
$570$	0	0
$571$	−19164.5	−1.40457	−0.702284	−	0.711897i	$-0.747836\pi$
$571$	−19164.5	−1.40457	−0.702284	+	0.711897i	$0.747836\pi$
$572$	−16460.9	−1.20326
$573$	0	0
$574$	−24088.3	−1.75161
$575$	5756.55	0.417504
$576$	0	0
$577$	8729.71	0.629848	0.314924	−	0.949117i	$-0.398021\pi$
$577$	8729.71	0.629848	0.314924	+	0.949117i	$0.398021\pi$
$578$	1226.12	0.0882353
$579$	0	0
$580$	−40498.2	−2.89930
$581$	−5798.99	−0.414084
$582$	0	0
$583$	18793.3	1.33506
$584$	3250.61	0.230328
$585$	0	0
$586$	12189.3	0.859279
$587$	−2927.87	−0.205871	−0.102935	−	0.994688i	$-0.532823\pi$
$587$	−2927.87	−0.205871	−0.102935	+	0.994688i	$0.532823\pi$
$588$	0	0
$589$	12735.9	0.890958
$590$	10982.7	0.766359
$591$	0	0
$592$	10011.3	0.695035
$593$	−2154.97	−0.149231	−0.0746157	−	0.997212i	$-0.523773\pi$
$593$	−2154.97	−0.149231	−0.0746157	+	0.997212i	$0.523773\pi$
$594$	0	0
$595$	3080.50	0.212249
$596$	16796.2	1.15436
$597$	0	0
$598$	11464.2	0.783958
$599$	−7065.12	−0.481925	−0.240963	−	0.970534i	$-0.577463\pi$
$599$	−7065.12	−0.481925	−0.240963	+	0.970534i	$0.577463\pi$
$600$	0	0
$601$	10656.5	0.723272	0.361636	−	0.932319i	$-0.382218\pi$
$601$	10656.5	0.723272	0.361636	+	0.932319i	$0.382218\pi$
$602$	−8743.20	−0.591937
$603$	0	0
$604$	6443.53	0.434079
$605$	16290.5	1.09471
$606$	0	0
$607$	82.8667	0.00554111	0.00277056	−	0.999996i	$-0.499118\pi$
$607$	82.8667	0.00554111	0.00277056	+	0.999996i	$0.499118\pi$
$608$	−13970.7	−0.931889
$609$	0	0
$610$	51083.8	3.39069
$611$	−5240.55	−0.346988
$612$	0	0
$613$	−15588.4	−1.02710	−0.513549	−	0.858060i	$-0.671670\pi$
$613$	−15588.4	−1.02710	−0.513549	+	0.858060i	$0.671670\pi$
$614$	1350.15	0.0887422
$615$	0	0
$616$	−5499.70	−0.359723
$617$	4430.58	0.289090	0.144545	−	0.989498i	$-0.453828\pi$
$617$	4430.58	0.289090	0.144545	+	0.989498i	$0.453828\pi$
$618$	0	0
$619$	3111.78	0.202056	0.101028	−	0.994884i	$-0.467787\pi$
$619$	3111.78	0.202056	0.101028	+	0.994884i	$0.467787\pi$
$620$	32419.9	2.10003
$621$	0	0
$622$	8230.95	0.530597
$623$	15735.2	1.01190
$624$	0	0
$625$	−19472.3	−1.24623
$626$	−23282.9	−1.48654
$627$	0	0
$628$	21305.3	1.35378
$629$	−3867.99	−0.245194
$630$	0	0
$631$	−14808.2	−0.934236	−0.467118	−	0.884195i	$-0.654708\pi$
$631$	−14808.2	−0.934236	−0.467118	+	0.884195i	$0.654708\pi$
$632$	2155.26	0.135651
$633$	0	0
$634$	−5673.66	−0.355410
$635$	−36971.8	−2.31052
$636$	0	0
$637$	−5756.94	−0.358082
$638$	61457.1	3.81365
$639$	0	0
$640$	−14699.5	−0.907887
$641$	9233.91	0.568982	0.284491	−	0.958679i	$-0.408175\pi$
$641$	9233.91	0.568982	0.284491	+	0.958679i	$0.408175\pi$
$642$	0	0
$643$	11712.7	0.718355	0.359177	−	0.933269i	$-0.383057\pi$
$643$	11712.7	0.718355	0.359177	+	0.933269i	$0.383057\pi$
$644$	10639.7	0.651028
$645$	0	0
$646$	3958.38	0.241084
$647$	12099.6	0.735216	0.367608	−	0.929981i	$-0.380177\pi$
$647$	12099.6	0.735216	0.367608	+	0.929981i	$0.380177\pi$
$648$	0	0
$649$	−9259.22	−0.560025
$650$	9807.70	0.591830
$651$	0	0
$652$	35824.1	2.15181
$653$	8335.37	0.499523	0.249761	−	0.968307i	$-0.419648\pi$
$653$	8335.37	0.499523	0.249761	+	0.968307i	$0.419648\pi$
$654$	0	0
$655$	27440.2	1.63691
$656$	19260.3	1.14633
$657$	0	0
$658$	−8754.54	−0.518674
$659$	−7751.68	−0.458213	−0.229107	−	0.973401i	$-0.573580\pi$
$659$	−7751.68	−0.458213	−0.229107	+	0.973401i	$0.573580\pi$
$660$	0	0
$661$	13808.0	0.812510	0.406255	−	0.913760i	$-0.366834\pi$
$661$	13808.0	0.812510	0.406255	+	0.913760i	$0.366834\pi$
$662$	−20476.9	−1.20220
$663$	0	0
$664$	−3793.67	−0.221721
$665$	9945.00	0.579925
$666$	0	0
$667$	−23778.9	−1.38039
$668$	18617.9	1.07837
$669$	0	0
$670$	−10581.8	−0.610167
$671$	−43067.2	−2.47778
$672$	0	0
$673$	−8406.44	−0.481493	−0.240746	−	0.970588i	$-0.577392\pi$
$673$	−8406.44	−0.481493	−0.240746	+	0.970588i	$0.577392\pi$
$674$	−35066.3	−2.00401
$675$	0	0
$676$	−11118.8	−0.632614
$677$	19257.9	1.09327	0.546633	−	0.837372i	$-0.315909\pi$
$677$	19257.9	1.09327	0.546633	+	0.837372i	$0.315909\pi$
$678$	0	0
$679$	3779.40	0.213608
$680$	2015.25	0.113649
$681$	0	0
$682$	−49198.1	−2.76231
$683$	3451.93	0.193388	0.0966942	−	0.995314i	$-0.469173\pi$
$683$	3451.93	0.193388	0.0966942	+	0.995314i	$0.469173\pi$
$684$	0	0
$685$	29102.7	1.62330
$686$	−28492.3	−1.58577
$687$	0	0
$688$	6990.82	0.387387
$689$	−12388.7	−0.685012
$690$	0	0
$691$	−26090.5	−1.43637	−0.718184	−	0.695853i	$-0.755026\pi$
$691$	−26090.5	−1.43637	−0.718184	+	0.695853i	$0.755026\pi$
$692$	−26125.0	−1.43515
$693$	0	0
$694$	−24782.7	−1.35553
$695$	23969.8	1.30824
$696$	0	0
$697$	−7441.50	−0.404400
$698$	−16432.2	−0.891072
$699$	0	0
$700$	9102.30	0.491478
$701$	−22283.9	−1.20064	−0.600321	−	0.799759i	$-0.704961\pi$
$701$	−22283.9	−1.20064	−0.600321	+	0.799759i	$0.704961\pi$
$702$	0	0
$703$	−12487.3	−0.669940
$704$	36378.6	1.94754
$705$	0	0
$706$	−17056.6	−0.909256
$707$	−20344.7	−1.08224
$708$	0	0
$709$	−4561.83	−0.241640	−0.120820	−	0.992674i	$-0.538552\pi$
$709$	−4561.83	−0.241640	−0.120820	+	0.992674i	$0.538552\pi$
$710$	68849.7	3.63927
$711$	0	0
$712$	10293.9	0.541825
$713$	19035.7	0.999847
$714$	0	0
$715$	−22996.8	−1.20284
$716$	−1266.46	−0.0661031
$717$	0	0
$718$	9639.52	0.501036
$719$	−12458.7	−0.646218	−0.323109	−	0.946362i	$-0.604728\pi$
$719$	−12458.7	−0.646218	−0.323109	+	0.946362i	$0.604728\pi$
$720$	0	0
$721$	5352.26	0.276461
$722$	−16321.2	−0.841290
$723$	0	0
$724$	17832.6	0.915393
$725$	−20343.0	−1.04209
$726$	0	0
$727$	19361.4	0.987721	0.493860	−	0.869541i	$-0.335585\pi$
$727$	19361.4	0.987721	0.493860	+	0.869541i	$0.335585\pi$
$728$	3625.46	0.184572
$729$	0	0
$730$	22706.4	1.15124
$731$	−2701.00	−0.136662
$732$	0	0
$733$	21638.4	1.09036	0.545180	−	0.838319i	$-0.316461\pi$
$733$	21638.4	1.09036	0.545180	+	0.838319i	$0.316461\pi$
$734$	31198.6	1.56889
$735$	0	0
$736$	−20881.3	−1.04578
$737$	8921.24	0.445886
$738$	0	0
$739$	−33520.7	−1.66858	−0.834290	−	0.551326i	$-0.814122\pi$
$739$	−33520.7	−1.66858	−0.834290	+	0.551326i	$0.814122\pi$
$740$	−31787.1	−1.57908
$741$	0	0
$742$	−20695.9	−1.02395
$743$	28486.4	1.40655	0.703274	−	0.710919i	$-0.251721\pi$
$743$	28486.4	1.40655	0.703274	+	0.710919i	$0.251721\pi$
$744$	0	0
$745$	23465.2	1.15396
$746$	1360.62	0.0667773
$747$	0	0
$748$	−8495.00	−0.415251
$749$	3104.92	0.151470
$750$	0	0
$751$	−29427.4	−1.42985	−0.714927	−	0.699199i	$-0.753540\pi$
$751$	−29427.4	−1.42985	−0.714927	+	0.699199i	$0.753540\pi$
$752$	6999.89	0.339441
$753$	0	0
$754$	−40513.2	−1.95677
$755$	9001.98	0.433928
$756$	0	0
$757$	30790.8	1.47835	0.739174	−	0.673514i	$-0.235216\pi$
$757$	30790.8	1.47835	0.739174	+	0.673514i	$0.235216\pi$
$758$	−2973.68	−0.142492
$759$	0	0
$760$	6505.97	0.310522
$761$	17677.5	0.842062	0.421031	−	0.907046i	$-0.361668\pi$
$761$	17677.5	0.842062	0.421031	+	0.907046i	$0.361668\pi$
$762$	0	0
$763$	−9853.05	−0.467502
$764$	−21444.4	−1.01548
$765$	0	0
$766$	−34865.9	−1.64459
$767$	6103.78	0.287346
$768$	0	0
$769$	6309.56	0.295876	0.147938	−	0.988997i	$-0.452736\pi$
$769$	6309.56	0.295876	0.147938	+	0.988997i	$0.452736\pi$
$770$	−38417.0	−1.79799
$771$	0	0
$772$	32058.2	1.49456
$773$	−10323.3	−0.480343	−0.240171	−	0.970731i	$-0.577204\pi$
$773$	−10323.3	−0.480343	−0.240171	+	0.970731i	$0.577204\pi$
$774$	0	0
$775$	16285.1	0.754811
$776$	2472.46	0.114377
$777$	0	0
$778$	16124.5	0.743049
$779$	−24023.9	−1.10494
$780$	0	0
$781$	−58045.2	−2.65944
$782$	5916.36	0.270548
$783$	0	0
$784$	7689.64	0.350293
$785$	29764.7	1.35331
$786$	0	0
$787$	11118.3	0.503589	0.251795	−	0.967781i	$-0.418979\pi$
$787$	11118.3	0.503589	0.251795	+	0.967781i	$0.418979\pi$
$788$	−27683.2	−1.25149
$789$	0	0
$790$	15055.1	0.678021
$791$	3792.37	0.170469
$792$	0	0
$793$	28390.4	1.27134
$794$	−25266.5	−1.12931
$795$	0	0
$796$	−7127.92	−0.317390
$797$	−32556.2	−1.44692	−0.723462	−	0.690364i	$-0.757450\pi$
$797$	−32556.2	−1.44692	−0.723462	+	0.690364i	$0.757450\pi$
$798$	0	0
$799$	−2704.50	−0.119748
$800$	−17864.1	−0.789487
$801$	0	0
$802$	−2469.08	−0.108711
$803$	−19143.1	−0.841279
$804$	0	0
$805$	14864.2	0.650802
$806$	32431.9	1.41733
$807$	0	0
$808$	−13309.4	−0.579484
$809$	−39644.3	−1.72289	−0.861445	−	0.507851i	$-0.830440\pi$
$809$	−39644.3	−1.72289	−0.861445	+	0.507851i	$0.830440\pi$
$810$	0	0
$811$	−7839.62	−0.339440	−0.169720	−	0.985492i	$-0.554286\pi$
$811$	−7839.62	−0.339440	−0.169720	+	0.985492i	$0.554286\pi$
$812$	−37599.4	−1.62497
$813$	0	0
$814$	48237.8	2.07707
$815$	50048.3	2.15106
$816$	0	0
$817$	−8719.82	−0.373400
$818$	−48186.6	−2.05967
$819$	0	0
$820$	−61154.1	−2.60438
$821$	−344.345	−0.0146379	−0.00731895	−	0.999973i	$-0.502330\pi$
$821$	−344.345	−0.0146379	−0.00731895	+	0.999973i	$0.502330\pi$
$822$	0	0
$823$	−43172.9	−1.82857	−0.914284	−	0.405074i	$-0.867246\pi$
$823$	−43172.9	−1.82857	−0.914284	+	0.405074i	$0.867246\pi$
$824$	3501.42	0.148031
$825$	0	0
$826$	10196.6	0.429522
$827$	21158.3	0.889656	0.444828	−	0.895616i	$-0.353265\pi$
$827$	21158.3	0.889656	0.444828	+	0.895616i	$0.353265\pi$
$828$	0	0
$829$	−10514.0	−0.440490	−0.220245	−	0.975445i	$-0.570686\pi$
$829$	−10514.0	−0.440490	−0.220245	+	0.975445i	$0.570686\pi$
$830$	−26499.9	−1.10822
$831$	0	0
$832$	−23981.1	−0.999275
$833$	−2971.00	−0.123576
$834$	0	0
$835$	26010.3	1.07799
$836$	−27425.0	−1.13458
$837$	0	0
$838$	88.8493	0.00366259
$839$	10036.2	0.412978	0.206489	−	0.978449i	$-0.433796\pi$
$839$	10036.2	0.412978	0.206489	+	0.978449i	$0.433796\pi$
$840$	0	0
$841$	59642.7	2.44548
$842$	18902.1	0.773647
$843$	0	0
$844$	7875.91	0.321209
$845$	−15533.6	−0.632394
$846$	0	0
$847$	15124.4	0.613555
$848$	16547.8	0.670113
$849$	0	0
$850$	5061.48	0.204244
$851$	−18664.1	−0.751817
$852$	0	0
$853$	9343.14	0.375033	0.187516	−	0.982261i	$-0.439956\pi$
$853$	9343.14	0.375033	0.187516	+	0.982261i	$0.439956\pi$
$854$	47427.2	1.90038
$855$	0	0
$856$	2031.22	0.0811048
$857$	−25235.0	−1.00585	−0.502923	−	0.864331i	$-0.667742\pi$
$857$	−25235.0	−1.00585	−0.502923	+	0.864331i	$0.667742\pi$
$858$	0	0
$859$	−39717.7	−1.57759	−0.788795	−	0.614656i	$-0.789295\pi$
$859$	−39717.7	−1.57759	−0.788795	+	0.614656i	$0.789295\pi$
$860$	−22196.7	−0.880119
$861$	0	0
$862$	10229.0	0.404176
$863$	26812.4	1.05760	0.528798	−	0.848748i	$-0.322643\pi$
$863$	26812.4	1.05760	0.528798	+	0.848748i	$0.322643\pi$
$864$	0	0
$865$	−36498.0	−1.43465
$866$	−7016.10	−0.275308
$867$	0	0
$868$	30099.3	1.17700
$869$	−12692.5	−0.495471
$870$	0	0
$871$	−5880.97	−0.228782
$872$	−6445.82	−0.250324
$873$	0	0
$874$	19100.2	0.739215
$875$	−9934.33	−0.383819
$876$	0	0
$877$	−33850.8	−1.30338	−0.651688	−	0.758487i	$-0.725939\pi$
$877$	−33850.8	−1.30338	−0.651688	+	0.758487i	$0.725939\pi$
$878$	63013.9	2.42211
$879$	0	0
$880$	30717.1	1.17668
$881$	19939.4	0.762514	0.381257	−	0.924469i	$-0.375491\pi$
$881$	19939.4	0.762514	0.381257	+	0.924469i	$0.375491\pi$
$882$	0	0
$883$	31421.5	1.19753	0.598765	−	0.800925i	$-0.295658\pi$
$883$	31421.5	1.19753	0.598765	+	0.800925i	$0.295658\pi$
$884$	5599.99	0.213063
$885$	0	0
$886$	22881.7	0.867635
$887$	−14713.6	−0.556971	−0.278486	−	0.960440i	$-0.589832\pi$
$887$	−14713.6	−0.556971	−0.278486	+	0.960440i	$0.589832\pi$
$888$	0	0
$889$	−34325.4	−1.29498
$890$	71905.7	2.70818
$891$	0	0
$892$	−29265.3	−1.09851
$893$	−8731.12	−0.327185
$894$	0	0
$895$	−1769.31	−0.0660801
$896$	−13647.3	−0.508844
$897$	0	0
$898$	−26070.3	−0.968793
$899$	−67269.7	−2.49563
$900$	0	0
$901$	−6393.49	−0.236402
$902$	92803.0	3.42572
$903$	0	0
$904$	2480.95	0.0912778
$905$	24913.2	0.915075
$906$	0	0
$907$	36405.1	1.33276	0.666380	−	0.745612i	$-0.267843\pi$
$907$	36405.1	1.33276	0.666380	+	0.745612i	$0.267843\pi$
$908$	62120.3	2.27041
$909$	0	0
$910$	25324.9	0.922540
$911$	−1574.12	−0.0572481	−0.0286241	−	0.999590i	$-0.509113\pi$
$911$	−1574.12	−0.0572481	−0.0286241	+	0.999590i	$0.509113\pi$
$912$	0	0
$913$	22341.3	0.809844
$914$	25630.6	0.927553
$915$	0	0
$916$	−45163.5	−1.62909
$917$	25476.1	0.917442
$918$	0	0
$919$	44823.7	1.60892	0.804460	−	0.594007i	$-0.202455\pi$
$919$	44823.7	1.60892	0.804460	+	0.594007i	$0.202455\pi$
$920$	9724.11	0.348472
$921$	0	0
$922$	−58674.9	−2.09583
$923$	38264.0	1.36455
$924$	0	0
$925$	−15967.2	−0.567566
$926$	53394.9	1.89489
$927$	0	0
$928$	73792.0	2.61028
$929$	10654.7	0.376287	0.188143	−	0.982142i	$-0.439753\pi$
$929$	10654.7	0.376287	0.188143	+	0.982142i	$0.439753\pi$
$930$	0	0
$931$	−9591.47	−0.337645
$932$	−35472.6	−1.24672
$933$	0	0
$934$	27839.8	0.975318
$935$	−11868.0	−0.415107
$936$	0	0
$937$	46738.5	1.62954	0.814770	−	0.579784i	$-0.196863\pi$
$937$	46738.5	1.62954	0.814770	+	0.579784i	$0.196863\pi$
$938$	−9824.40	−0.341981
$939$	0	0
$940$	−22225.5	−0.771188
$941$	31346.7	1.08594	0.542972	−	0.839751i	$-0.317299\pi$
$941$	31346.7	1.08594	0.542972	+	0.839751i	$0.317299\pi$
$942$	0	0
$943$	−35907.2	−1.23998
$944$	−8152.91	−0.281096
$945$	0	0
$946$	33684.2	1.15768
$947$	18191.9	0.624242	0.312121	−	0.950042i	$-0.398961\pi$
$947$	18191.9	0.624242	0.312121	+	0.950042i	$0.398961\pi$
$948$	0	0
$949$	12619.4	0.431656
$950$	16340.3	0.558053
$951$	0	0
$952$	1871.00	0.0636969
$953$	−47969.6	−1.63052	−0.815261	−	0.579094i	$-0.803407\pi$
$953$	−47969.6	−1.63052	−0.815261	+	0.579094i	$0.803407\pi$
$954$	0	0
$955$	−29959.0	−1.01513
$956$	−7269.69	−0.245940
$957$	0	0
$958$	70467.8	2.37653
$959$	27019.6	0.909810
$960$	0	0
$961$	24060.3	0.807637
$962$	−31798.9	−1.06573
$963$	0	0
$964$	16896.7	0.564529
$965$	44787.1	1.49404
$966$	0	0
$967$	51466.6	1.71154	0.855768	−	0.517360i	$-0.173085\pi$
$967$	51466.6	1.71154	0.855768	+	0.517360i	$0.173085\pi$
$968$	9894.32	0.328528
$969$	0	0
$970$	17270.9	0.571684
$971$	4007.62	0.132452	0.0662259	−	0.997805i	$-0.478904\pi$
$971$	4007.62	0.132452	0.0662259	+	0.997805i	$0.478904\pi$
$972$	0	0
$973$	22254.0	0.733229
$974$	−33938.8	−1.11650
$975$	0	0
$976$	−37921.5	−1.24369
$977$	37362.2	1.22346	0.611731	−	0.791066i	$-0.290474\pi$
$977$	37362.2	1.22346	0.611731	+	0.791066i	$0.290474\pi$
$978$	0	0
$979$	−60621.6	−1.97903
$980$	−24415.6	−0.795844
$981$	0	0
$982$	49508.0	1.60882
$983$	38659.1	1.25436	0.627179	−	0.778875i	$-0.284210\pi$
$983$	38659.1	1.25436	0.627179	+	0.778875i	$0.284210\pi$
$984$	0	0
$985$	−38675.0	−1.25106
$986$	−20907.7	−0.675292
$987$	0	0
$988$	18078.8	0.582150
$989$	−13033.0	−0.419035
$990$	0	0
$991$	46.8701	0.00150240	0.000751200	−	1.00000i	$-0.499761\pi$
$991$	46.8701	0.00150240	0.000751200		1.00000i	$0.499761\pi$
$992$	−59072.5	−1.89068
$993$	0	0
$994$	63921.5	2.03971
$995$	−9958.11	−0.317280
$996$	0	0
$997$	−16913.4	−0.537265	−0.268633	−	0.963243i	$-0.586572\pi$
$997$	−16913.4	−0.537265	−0.268633	+	0.963243i	$0.586572\pi$
$998$	−88655.2	−2.81196
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.e.1.2		2
3.2	odd	2		51.4.a.d.1.1	✓	2
4.3	odd	2		2448.4.a.v.1.2		2
12.11	even	2		816.4.a.o.1.1		2
15.14	odd	2		1275.4.a.m.1.2		2
21.20	even	2		2499.4.a.l.1.1		2
51.50	odd	2		867.4.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.d.1.1	✓	2	3.2	odd	2
153.4.a.e.1.2		2	1.1	even	1	trivial
816.4.a.o.1.1		2	12.11	even	2
867.4.a.j.1.1		2	51.50	odd	2
1275.4.a.m.1.2		2	15.14	odd	2
2448.4.a.v.1.2		2	4.3	odd	2
2499.4.a.l.1.1		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.a (trivial)

\(f(q)\)	\(=\)	\(q+4.24264 q^{2} +10.0000 q^{4} +13.9706 q^{5} +12.9706 q^{7} +8.48528 q^{8} +59.2721 q^{10} -49.9706 q^{11} +32.9411 q^{13} +55.0294 q^{14} -44.0000 q^{16} +17.0000 q^{17} +54.8823 q^{19} +139.706 q^{20} -212.007 q^{22} +82.0294 q^{23} +70.1766 q^{25} +139.757 q^{26} +129.706 q^{28} -289.882 q^{29} +232.059 q^{31} -254.558 q^{32} +72.1249 q^{34} +181.206 q^{35} -227.529 q^{37} +232.846 q^{38} +118.544 q^{40} -437.735 q^{41} -158.882 q^{43} -499.706 q^{44} +348.021 q^{46} -159.088 q^{47} -174.765 q^{49} +297.734 q^{50} +329.411 q^{52} -376.087 q^{53} -698.117 q^{55} +110.059 q^{56} -1229.87 q^{58} +185.294 q^{59} +861.852 q^{61} +984.542 q^{62} -728.000 q^{64} +460.206 q^{65} -178.530 q^{67} +170.000 q^{68} +768.792 q^{70} +1161.59 q^{71} +383.088 q^{73} -965.324 q^{74} +548.823 q^{76} -648.146 q^{77} +254.000 q^{79} -614.705 q^{80} -1857.15 q^{82} -447.088 q^{83} +237.500 q^{85} -674.080 q^{86} -424.014 q^{88} +1213.15 q^{89} +427.265 q^{91} +820.294 q^{92} -674.955 q^{94} +766.736 q^{95} +291.383 q^{97} -741.463 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{4} - 6 q^{5} - 8 q^{7} + 144 q^{10} - 66 q^{11} - 2 q^{13} + 144 q^{14} - 88 q^{16} + 34 q^{17} - 26 q^{19} - 60 q^{20} - 144 q^{22} + 198 q^{23} + 344 q^{25} + 288 q^{26} - 80 q^{28} - 444 q^{29}+ \cdots - 1152 q^{98}+O(q^{100}) \) 2 * q + 20 * q^4 - 6 * q^5 - 8 * q^7 + 144 * q^10 - 66 * q^11 - 2 * q^13 + 144 * q^14 - 88 * q^16 + 34 * q^17 - 26 * q^19 - 60 * q^20 - 144 * q^22 + 198 * q^23 + 344 * q^25 + 288 * q^26 - 80 * q^28 - 444 * q^29 + 532 * q^31 + 600 * q^35 + 88 * q^37 + 576 * q^38 + 288 * q^40 - 570 * q^41 - 182 * q^43 - 660 * q^44 - 144 * q^46 - 420 * q^47 - 78 * q^49 - 864 * q^50 - 20 * q^52 + 300 * q^53 - 378 * q^55 + 288 * q^56 - 576 * q^58 - 444 * q^59 + 400 * q^61 - 288 * q^62 - 1456 * q^64 + 1158 * q^65 - 968 * q^67 + 340 * q^68 - 1008 * q^70 + 1848 * q^71 + 868 * q^73 - 2304 * q^74 - 260 * q^76 - 312 * q^77 + 508 * q^79 + 264 * q^80 - 1296 * q^82 - 996 * q^83 - 102 * q^85 - 576 * q^86 - 288 * q^88 + 288 * q^89 + 1160 * q^91 + 1980 * q^92 + 432 * q^94 + 2382 * q^95 + 1024 * q^97 - 1152 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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